(Note: I might have screwed up orientations, so take everything with a grain of salt.) I will write an argument for the case $m=1$, showing that the reduced contact invariant $c^{\rm red}(\xi)$ is non-zero. I think that the argument(s) can be tweaked to work for $m>1$, as well, but I won't try to write it down properly.
Let me suppress some things from the notation: I'll call $\Sigma = \Sigma(2,3,7)$, and let $\xi$ denote either of $\xi_0$ or $\xi_1$. (The statement is invariant under self-diffeomorphisms of $\Sigma$.)
Mark and Tosun give an explicit surgery presentation of $\xi$ as Legendrian surgery along a right-handed trefoil with Thurston-Bennequin $0$. This gives a Stein cobordism $W: S^3 \leadsto \Sigma$, and the cobordism map $F: = F^+_{\overline W}: HF^+(-\Sigma) \to HF^+(-S^3)$ maps $c(\xi)$ to $c(\xi_{\rm std})$.
The key point is showing that $F$ vanishes on the tower in $HF^+(\Sigma)$ (i.e. it vanishes for sufficiently large degrees). This can be seen in two ways.
The first is to show that spin$^c$ structures on $W$ come in conjugate pairs, and all their contributions cancel in pairs on the tower. (Note that they can -and do, a posteriori- move elements outside the tower.)
The second is that $F$ fits into an exact triangle:
$$ \dots \to HF^+(-\Sigma_0) \to HF^+(-\Sigma) \to HF^+(-S^3)\to \dots$$
where $\Sigma_0$ is the 0-surgery along the (right-handed) trefoil.
It's easy to see (basically by rank count) that the map $HF^+(-\Sigma_0) \to HF^+(-\Sigma)$ must be surjective onto the tower, which must then be killed by $F$.
Either way, if $c^{\rm red}(\xi)=0$, then $F(c^{\rm red}(\xi)) = 0 \neq c(\xi_{\rm std}))$.